Contents

World records

[math]H[/math] is a quantity such that there are infinitely many pairs of consecutive primes of distance at most [math]H[/math] apart. Would like to be as small as possible (this is a primary goal of the Polymath8 project).

[math]k_0[/math] is a quantity such that every admissible [math]k_0[/math]-tuple has infinitely many translates which each contain at least two primes. Would like to be as small as possible. Improvements in [math]k_0[/math] lead to improvements in [math]H[/math]. (The relationship is roughly of the form [math]H \sim k_0 \log k_0[/math]; see the page on finding narrow admissible tuples.)

[math]\varpi[/math] is a technical parameter related to a specialized form of the Elliott-Halberstam conjecture. Would like to be as large as possible. Improvements in [math]\varpi[/math] lead to improvements in [math]k_0[/math], as described in the page on Dickson-Hardy-Littlewood theorems. In more recent work, the single parameter [math]\varpi[/math] is replaced by a pair [math](\varpi,\delta)[/math] (in previous work we had [math]\delta=\varpi[/math]). These estimates are obtained in turn from Type I, Type II, and Type III estimates, as described at the page on distribution of primes in smooth moduli.

In this table, infinitesimal losses in [math]\delta,\varpi[/math] are ignored.

[math]k_0[/math] bound uses the optimal Bessel function cutoff. Originally only provisional due to neglect of the kappa error, but then it was confirmed that the kappa error was within the allowed tolerance.

Systematic tables of tuples of small length have been set up here and here (update: As of June 27 these tables have been merged and uploaded to an online database of current bounds on [math]H(k)[/math] for [math]k[/math] up to 5000).

Page 31: The estimation of (5.14) by (5.15) does not appear to be justified. In the text, it is claimed that the second summation in (5.14) can be treated by a variant of (4.15); however, whereas (5.14) contains a factor of [math](\log \frac{R}{|D|})^{2\ell+1}[/math], (4.15) contains instead a factor of [math](\log \frac{R/w}{|K|})^{2\ell+1}[/math] which is significantly smaller (K in (4.15) plays a similar role to D in (5.14)). As such, the crucial gain of [math]\exp(-k\omega/3)[/math] in (4.15) does not seem to be available for estimating the second sum in (5.14).

Page 7: In (2.39), the exponent of [math]3a/2[/math] should instead be [math]-5a/2[/math] (it comes from dividing (2.38) by (2.37)). This impacts the numerics for the rest of the paper.

Page 8: The "easy calculation" that the relative error caused by discarding all but the smooth moduli appears to be unjustified, as it relies on the treatment of (5.14) in Motohashi-Pintz which has the issue pointed out in 2.1 above.

[R1974] Richards, Ian On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem. Bull. Amer. Math. Soc. 80 (1974), 419–438. MathSciNetArticle